Lima, Peru is 10 degrees south of the equator. How many miles is that?

I believe one minute of latitude = one nautical mile
Then 10 degrees = 600 minutes = 600 NM
600 NM converts to 690.5 statue miles south of the equator

Can anyone confirm this?

That is roughly correct and is based on approximating the earth as sphere. It is actually a slightly oblate ellipsoid, so the length of a minute of latitude is slightly different at the poles and equator (varies smoothly in between).

Navigation texts such as the American Practical Navigator have tables that detail this more accurately. However, for estimates, the method you used is very common and often sufficiently accurate, good to about 1/2%.